Emit Hair Particle from Center of Face
To make particles emitted evenly from each face's center set up the next things:


Uncheck Random in Particle System settings (which you already did, still for completeness).
Make sure the object which will be used as particle instance has origin in the center of its geometry.
If you don't this still will work but when scaling particles' size it will grow somewhat unexpected.
Set Particles/Face to 1 in Particle System settings (by default 0, i.e. automatic selection). This will mean that you ask particle system to emit only 1 particle per emitter's face.





This will work in a similar way for Emitter type of Particle System too.
Number of particles - in order all this to work it has to be not less than total number of faces of the emitter. It may be higher though (doesn't matter how much, Particle System will automaticallsy cut what's not needed).
This behaviour is less predictable if Even Distribution in Particle System settings is left being turned on. In order to make number of particles exactly fit amount of faces uncheck this option.